In this paper, we also assume that \(({X}\) , \({Y})\)  is an admissible balanced pair in \(\mathrm {Mod}\)\(R\) .
Whenever the subcategory \({X}\)  (respectively, \({Y}\) ) appears, we always suppose
that it is a part of the admissible balanced pair \(({X}\) , \({Y})\) .
Note that the functor \(\mathrm {Ext}^{i}_{{A}}(-,~-)\)  is based on the classical balanced pair \((\mathrm {Proj} R,~\mathrm {Inj} R)\) , it induces
an isomorphism of cohomology groups whether we take a projective resolution of the first variable or take
an injective coresolution of the second variable. From this viewpoint, we have the functor \(\mathrm {Ext}^{i}_{{X}}(-,~-)\)  with respect
to admissible balanced pair \(({X}\) , \({Y})\) . By Lemma 2.1 [1]}, the functor is well-define.
